Analytic structures in the maximal ideal space of a uniform algebra

A R K I V F O R M A T E M A T I K B a n d 8 n r 23 1.70 11 1 Communicated 11 September 1970 by L E ~ R ~ C~m~F.SO~¢

Analytic structures in the maximal ideal space o f a uniform algebra

B y JAr~-EIuK B J 6 R K

Introduction

L e t A be a uniform algebra with its maximal ideal space MA and its Shilov boundary S~. We say t h a t M~ has a (one-dimensional) analytic structure at a point x o EM~'~S.4 if the following condition holds. There is an open neighborhood W of x o in M~ and s o m e / C A such t h a t W'~{xo} = V 1 U ... U V~, where V~ are disjoint open subsets of M~ each mapped homeomorphicalty b y ~ onto the set D~.{0} in C ~. Here D is the open unit disc and f(xo)=-0. The positive integer n above is called the branch-order of x o.

If MA has an analytic structure at the point x 0 as above, then J. Wermer has proved t h a t if g e g and if we define g~(z)=g(x~(z)) on D ~ { 0 } , where xl(z) is the point in Vi for which ](xt(z)) =z while g,(0) =g(x0), then gl ... g~ are analytic functions in D.

Conditions which guarantee t h a t subsets of M A ~ S A have an analytic structure have originally been studied b y J. Wermer in [5-6]. In Section 1 of this paper we prove some results which originally were obtained b y Wermer under certain regularity conditions. The core of this section is the proof of Theorem 1.7. and in the final part we discuss some consequences of this result.

Section 1

Firstly we introduce some notations and collect some wellknown facts about uni- form algebras. If X is a compact space and if ] e C(X) we put ]/Iz = sup({/(x) [: x e X}.

If W is a subset of X then aW denotes its topological boundary. If A is a uniform algebra and if F is closed subset of M~ we p u t HullA(F ) = { x e M A : ]](x)l <~ I/IF for all f in A}. We also introduce the uniform algebra A ( F ) = { g e C ( F ) : 3(f,) in A with l i m l £ - g l F = 0 } . Here we know t h a t MA(F) can be identified with HUllA(F).

If A is a uniform algebra and if l e A we define the fibers g71(z)=(xeMA :/(x) =z}

for each z E C 1. We recall the wellknown result below.

L e m m a 1.1. Let A be a unilorm algebra and let / E A . Suppose that U is an open compo- nent o/the set C*".,](SA). Then two cases are possible, either U ~ /(MA) is empty or else U c ](MA).

N e x t we introduce a concept which appears in [2, p. 525].

j..~.., ~Jfir.K Analytic structures in the maximal ideal space of a uniform algebra Definition 1.2. Let A be a uni/orm algebra and let l E A . Let U be an open component o/ the set C a ' ~ ( S ~) such that the fibers ~$-l(z) contain at most n points/or each z E U, while equality holds/or some point. Here n >~O and we say that U is an/-regular component o/

multiplicity n.

I n [2] E. Bishop proves t h a t if U is a n / - r e g u l a r c o m p o n e n t of multiplicity n, with n > 0 , t h e n there exists n subsets W1 ... W~ in MA each m a p p e d h o m e o m o r p h i c a l l y b y / o n t o U. I f t h e n g E A a n d if we p u t g~(z) =g(gr-i(z) N W~) for all zE U, t h e n gi ... g~

are analytic in U. T h e r e also exists some g in A such t h a t gi... g. are distinct. So if we p u t D i j = ( z E U: g~(z)=g~(z)}, t h e n W~ N W~ is c o n t a i n e d in t h e discrete set z~s -i (D~j). This result shows t h a t t h e o p e n set gr-~(U) has analytic s t r u c t u r e a t all points a n d t h e b r a n c h - o r d e r is one except for a discrete subset where t h e b r a n c h - o r d e r varies between 2 a n d n.

N o w we wish t o state some criteria which g u a r a n t e e t h a t a n open c o m p o n e n t of t h e set Ca'x,](S~) i s / - r e g u l a r w h e n / belongs t o some A. F i r s t l y we h a v e t h e so called Local Principle which appears in [2, L e m m a 17].

L e m m a 1.3. I, et U be an open component o/the set Ca~,J(SA). Suppose that U contains an open subset V such that @ ~ - l ( z ) = n/or all z E V. Then U is/-regular o/ multiplicity n.

Notice t h a t L e m m a 1.1. is t h e case w h e n n = 0 in L e m m a 1.3. T h e n e x t result is t h e so-called A n a l y t i c Disc L e m m a .

L e m m a 1.4. L e t / E A and let D be a closed Jordan curve with its interior A. Suppose that D contains an open subarc J such that :~gr-i(z) <~n /or all z e J and that the (possibly empty) set g T i ( A ) is contained in M A~S.4. Then @~Fi(z) <~n /or all z in A.

Proo/. T h e case w h e n n = l is c o n t a i n e d in [6, L e m m a 8]. I f n>~2 we can choose zoEJ so t h a t = ~ g l - i ( z 0 ) = n say, while :~gr-i(z)~<n for all z in J N D(z0), where D(zo) is an o p e n disc centered a t z 0. L e t x 1 ... x , be t h e points in gFi(z0) a n d choose disjoint o p e n neighborhoods W i ... W. of these points so t h a t / ( W ~ ) A J is c o n t a i n e d in/)(z0) for each i.

N o w we can easily find a closed J o r d a n curve D 0 with its interior A 0 such t h a t I~o A F is a closed subarc of t h e set J A D(zo) while A 0 is contained in A. I n addition we can arrange this so t h a t ~1-i(A0 U D0) is contained in W1 U ... (J W~, since this set is an o p e n n e i g h b o r h o o d of t h e whole fiber ~r-i(z0).

l~rom t h e ease n = 1 we can conclude t h a t 7~r-i(z) n Wi contains a t m o s t one p o i n t f o r each i, a n d hence :~qzFi(z) ~<n for all zE/k o. H e r e A o is a n o p e n subset of t h e con- n e c t e d set A, so a n application of L e m m a 1.3. completes t h e proof.

Using L e m m a 1.4. a n d T h e o r e m 1.7. below we can easily deduce t h e following result.

Proposition 1.5. Let ] E A and suppose that/(S~) contains a relatively open subset J which is homeomorphic with an open Jordan arc. SupTose also that the sets S z = { x E S A :/(x) =z } contain at most n p o i n t s / o r each z in J. Suppose that U and V are the two (distinct) componsnts o/ the set Ca~,J(S4) which borders J, and that V is/-regular o/ multiplicity m. Here m = O may occur. Then it/oUows that U is /-regular o/ some multiplicity

<~m +n.

L e t us r e m a r k t h a t Proposition 1.5. above was p r o v e d b y J . W e r m e r in t h e case w h e n J is real analytic a n d it also appears in [2, L e m m a 21] w h e n n = 1 u n d e r a less

2 4 0

A . R K I Y FOR M~.TEMATIK. Bd 8 nr 23 restrictive regularity assumption about J . Finally we wish to point out t h a t Propo- sition 1.5. is contained in the work b y H. Alexander in [1].

Next we wish to state a result, again due to J. Wermer which will be used in the proof of Theorem 1.7. below. Firstly we recall t h a t a curve F in C 1 is real-analytic if there is a non-constant real-analytic function ~ defined on T such t h a t ~ ( T ) = F . If we assume t h a t the origin 0 belongs to C l e f we can introduce the winding number of F with respect to 0 in the usual way. I f F 1 and F~ are two real-analytic curves we say t h a t I~lc F2 if F 1 is contained in the open component of the set C a l F 2 which contains 0.

Proposition

holds and in addition F~ c (z E C ~ : I z J > 1 + e }. Let now N be the maximum of the win- ding number of 0 / o r each F~. I] U is the open component of the set cr"..J(SA) which contains the point 1, then U is/-regular o/some multiplicity <~ nN.

I n Theorem 1.7. below we apply Proposition 1.6. to obtain a general result dealing with so called crossing over edges. The reader will see t h a t we have taken great ad- vantage of [2] in the proof. I n particular L e m m a 1.8. was already proved there.

Theorem 1.7. Let A be a uniform algebra and let / 6 A . Let U be an/-regular component o/mult•licity n. Let Zo6OU be such that ~I-1(zo) ~ SA is a non-empty A-convex set , in M A. Then the set gr-l(Zo)~SA contains at most n points and in addition M A has ana- lytic structure at each point in this (possibly empty) set.

Proof. Let us assume t h a t n > 0 (the case when n = 0 is easy and contained in the following proof using Lemma 1.8.) and choose a point 4 o in U such t h a t ~71(A0) contains n points w 1 ... w~. Denote b y D o a small open disc centered at 4 o so t h a t gFX(D0) = W 1 U ... O W~, where W, are disjoint open subsets of MA each mapped homeomorphically b y / onto D 0.

Next e > 0 is so small that if D(e) is the open disc of radius e centered at 40, then D(e) U T(e) is contained in D 0. Here T(e) is the circle of radius s centered at )t 0. At this stage we do not fix e, the following lemmas will be true for a n y 8 as above, and in the final part of the proof we choose ~ to be sufficiently small.

Let us p u t X = M A ~ I - I ( D ( s ) ) , so t h a t X is a closed subset of MA. Here xi-l(T(e))

= a X and we know t h a t ~X = K 1 U ... U Kn, where K, are disjoint closed sets each mapped homeomorphically b y / onto T(s). Next we introduce the uniform algebra B on X which is generated by the restriction algebra A ]X and the function (/-40) -1 = /0 in C(X). I n the series of lemmas which follows we derive some properties of B which finally will enable us to prove Theorem 1.7.

Lemma 1.8. Let B and X be as above. Ther~ M~ = X and Ss is contained in the set (S,~UK~U ... UK,~).

Proof. L e t % e M s be given. Since B contains A ]X we get a point Yo in MA such t h a t g(xo)=g(Yo) for all g e A I X . Since (/-40) is invertible in B it follows that/(MB) does n o t contain )l o. I n addition ] ( S z ) c / ( X ) which means t h a t D ( e ) c G l " \ l ( S s ) . Then L e m m a 1.1. shows t h a t ](M~) N D(8) is e m p t y which implies t h a t yoEX. B u t

j..v. ~JSaz, Analytic s ~ r ~ r e s in the maximal ideal space of a uniform a~ebra n o w it is easily seen that/o(xo) •/o(Yo) which implies t h a t x o =yo in M s so t h a t M s = X follows.

Suppose now t h a t S~ is not contained in the set R -~ S A 0 K 1 [J ... (J K n. Then X ~ R contains a point y which is a p e a k point for B. Here y ~ X ~ S X so we can choose a closed A-convex neighborhood W of y in MA while W ~ X. H e r e / - 4 0 ~=0 on W which means t h a t ~o] W belongs to A(W). Hence A ( W ) = B ( W ) and if we let g ~ B peak at y, t h e n A contains a sequence (g.) such t h a t lim I g , - g I w =0" B u t then we s e e t h a t i f n is sufficiently large the function g, will determine a local peak set in the interior of W. Since W ~ M ~ S a this contradicts the Local M a x i m u m Principle. This proves t h a t S ~ R m u s t hold.

L e m m a 1.9. Hull~(Sa) does not intersect r~'~(U).

Proo I. L e t B1 ~B(HullA(SA) ). Then B 1 is a uniform algebra and Ms~=Hulls(SA) while SSl--S~. Consider / as an element of B1. Since Ms~ ~ X we see t h a t ~t 0 belongs to 5u~f(MB~) and in addition U ~ C-a~](Ss:). Hence an application of L e m m a 1.1.

proves t h a t / ( M s ~ ) N U is e m p t y which gives the desired result.

L e m m a 1.10. Let B 1 be as in Lemma 1.9. Then gr-l(Zo) N MB~ is a peak set/or B r Proo]. L e t us p u t Y--](Mm) so t h a t Y A U is e m p t y . Since U is connected and zoEaV there exists a function P E C( Y) such t h a t P(zo)= 1 while [P[ < 1 on Y~{z0}.

In addition P can be uniformly a p p r o x l m a t e d on Y b y rational functions R~ which h a v e poles in U only. Since YN U is e m p t y the functions R,,o/EB 1 and hence it follows t h a t P1 - - P o f belongs to B 1. C l e a r l y P 1 determines the peak set g[l(z0) N Ms~.

L e m m a 1.11. Let y Ercr-l(Zo)~S4. Then y does not belong to Hulls(SA).

Proo/. Suppose t h a t yEHulls(Sa) = M s , . Since g71(z0) N Ms1 is a p e a k set for B 1 and since S ~ - - - S a we can conclude t h a t y even belongs to Hull~(Sa N gTl(zo). I f K = H u l l ~ ( S a A gFl(zo)) it is obvious t h a t K is contained in M m N g71(z0), so the function ]o is constant on K. This implies t h a t BI(K ) = B ( K ) = A ( K ) and hence K = Hulla(S a N g71(zo)), so b y assumption K - ~ S a N ~71(zo). B u t here y E M a ~ S a so t h a t y E M a ~ K , a contradiction.

Gontinuation o/ the Troo/o/ Theorem 1.7. L e t yo E ~ 7 1 ( z 0 ) ~ S a be a given point.

Using L e m m a 1.11. we can choose g in B so t h a t IgiSA<~0< ½ while g ( y o ) = l . Clearly we m a y change g slightly so we m a y assume t h a t g=h/g N where h E A I X and N >10. Here N > 0 since g clearly cannot belong to A X. I n addition we m a y assume t h a t t h e function h is such t h a t 0 < h(wl) i < . . . < l h(w,)] and t h a t dh~/dz 4:0 a t ~0 for each i, where h~ are the analytic functions determined b y h on D 0.

A t this stage we m a k e a good choice of e. F o r we can choose e so small t h a t if Fi =g(Kt), t h e n F~ are analytic curves where F I = ... c F , and each Ft has the winding n u m b e r N with respect to 0. I n addition we m a y arrange this so t h a t F1c(zECl:

t z I > 1 + e0}. So now a n application of Proposition 1.6. shows t h a t M s has an ana- lytic structure at the point Y0.

H e r e Y0 E X ' ~ O X so we can choose a small closed neighborhood W of Yo in MA as in L e m m a 1.8. for which A ( W ) = B ( W ) . B u t t h e n it follows t h a t MA also has an analytic structure at Y0.

Finally we show w h y =~= ( g / " l ( z o ) ~ S A ) ~ n . F o r if y E T ~ / - I ( Z o ) ~ S A t h e n the fact t h a t M A has an analytic structure a t y implies t h a t if / is not locally constant in a 242

~ v FSR ~ t T E ~ n ~ . Bd 8 ar 23 neighborhood of y, then f(W) is a neighborhood of z o in C a if W is a neighborhood of y in M , . So if gr-:(zo)~S,~ contains n + 1 points Yl ... Y.+: where / is not locally constant, t h e n we choose disjoint neighborhoods W: ... W.+I of these points and now we can find an open disc D(zo) which is contained in I(W:)N ... N/(W,+:). Since zoEOU we can choose z:ED(zo) N U and then g/-x(zl) N Ws are non e m p t y for each i, so t h a t :~ gf-:(z:) >~n + 1, a contradiction.

Hence the set Q={yEg/-l(Zo)~S~: f is not locally constant at y}, contains at most n points. If we p u t Z = Q U (gj-:(z0) N S~), then it is clear t h a t z/-:(Zo)~Z is an open subset of M~. L e t us introduce the uniform algebra Al=A(gi-l(zo)). Using the Local Maximum Principle it follows t h a t S~1 is contained in Z. Since Q is a finite set it follows t h a t M~=gf-:(Zo) is contained in Q U HullA~(gr-l(z0)N S~)=

Q U Hull~ (g1-:(z0) N S~) =Q U (gr-:(z0) fi S~). This completes the proof of Theorem 1.7.

The n e x t result is fairly direct consequence of the preceeding proof.

Theorem 1.12. Let A be a uniform algebra and let / be a function in A such that each open component o/the set Ca~f(SA) is f-regular o/some multiplicity n >~O. I n addition we assume that R(f(SA)) =C(](S~)) and that the sets x~7:(z ) A ,.~A are A.convex in M.4 for all zEf(SA). Then it follows that M ~ S ~ has an analytic structure.

Proof. L e t z o E/(S~) and assume t h a t x is a point in g~-:(Zo)~S ~. Using the assump- tion t h a t R(f(S~))=C(/(SA)) the arguments used in L e m m a 1.9-11 show t h a t there exist finitely m a n y components U: ... U~ of the set Ca~f(S~) satisfying the following condition. If we take a point ~s in each Us and choose small open discs D(2~) and let B be the uniform algebra on the set X - M ~ / ( ~ : (D(2:) U ... U D()~)) which is generated b y A ] X and the functions ( / - 2 s ) - : , then x does not belong to Hulls(S~).

Using this fact the same argument as in the proof of Theorem 1.7. shows t h a t Ma has an analytic structure at the point x 0.

In this final part we derive some consequences of Theorem 1.7. and 1.12. above.

Firstly we notice t h a t Theorem 1.7. together with Proposition 1.5. immediately shows t h a t Proposition 1.6. holds in the case when Fs are curves with finitely m a n y intersection points, i.e. F~=/(K~) where K s contains a finite set Ss such t h a t ](x)4=

fly) for all pairs x . y in Ks~Ss.

Using the fact above we can for example derive the following result which was independently obtained b y H. Alexander in [1].

Theorem 1.13. Let J be a closed curve in Ca with finitely many intersection points and let A be a uniform algebra on J which contains the coordinate/unction z. Then the (possibly empty) set M A~S~ has an analytic structure.

Finally we show how the result of G. Stolzenberg in [6] can be derived from Theo- rem 1.12.

Theorem 1.14. Let A be a uniform algebra on the unit circle T such that A is generated by continuously differentiable functions. Then the (possibly empty) set M ~'~S~ has an analytic structure.

Before we begin the proof we insert some preliminary results. Suppose t h a t ] 6 Ca(T) and let E be a closed totally disconnected subset of T. Then C a \ , ] ( E ) is connected so if A is a uniform algebra on T which contains f, then an application of :Kerge- lyan's Theorem to the simply connected set ](E) shows t h a t if ~EO(E) t h e n c e / actually belongs to the closed restriction algebra A(E). I t follows t h a t if A is gene- rated b y continuously differentiable functions then A ( E ) = C(E).

J.-E. BJSR~:, Analygic structures in the maximal ideal space of a uniform algebra T h e n e x t r e s u l t is e s s e n t i a l l y c o n t a i n e d i n [ T h e o r e m , p. 509 i n 2], t h o u g h w e also n e e d P r o p o s i t i o n 1.5. a n d 1.6. w i t h a s m o o t h J o r d a n arc.

P r o p o s i t i o n 1.15. Let A be a uniform algebra on T and let f E CI( T) be an element of A.

Assume that T contains a closed totally disconnected subset E such that if x E T ~ E , then the set

S~=(yeT: /(y)=/(z)}

finite subset o/

Then it follows that each open component of the set C l e f ( T ) is/.regular.

Proo/ o/ Theorem 1.14. F i r s t l y i t is e a s i l y v e r i f i e d t h a t A c o n t a i n s s o m e / 6 C 1 ( T ) s a t i s f y i n g t h e c o n d i t i o n s i n P r o p o s i t i o n 1.15. w i t h r e s p e c t t o a closed t o t a l l y discon- n e c t e d s u b s e t E of T . C l e a r l y S ~ c T (in f a c t SA = T b y Corollary, p. 85 in 6), so using T h e o r e m 1.12. i t is sufficient t o s h o w t h a t R(f(T)) =C(/(T)) a n d t h a t ~r-l(zo) N T is A - c o n v e x for all %El(T). N o w f E C I ( T ) so t h e p l a n a r L e b e s g u e m e a s u r e of f ( T ) is zero a n d t h e n i t is welll~nown t h a t R(f(T))=C(f(T)). N e x t E(zo)=r~r-l(zo)N T is a l w a y s a closed d i s c o n n e c t e d s u b s e t of T w h e n z 0 E ](T), so b y t h e p r e c e e d i n g discussion A(E(zo) ) =C(E(zo) ) w h i c h m e a n s t h a t E(zo)=MA(s(~o~)=HullA(~s-l(z0) N T). T h i s

c o m p l e t e s t h e p r o o f .

Depar~men$ o/MaSh~,m~, University of Stockholm, Stockholm, Sweden

R E F E R E N C E S

1. Ar~XA~E~, H., Uniform algebras on curves. Bull Math. Soc. 75: 6, 1269-1272 (1969).

2. BISHOP, •., A~ialyticityin cert~inBanach algebras. Trans. Amer. Math. Soe. I02, 507-544 (1962) 3. BJSI~K, J.-E., Relatively maximal function algebras generated by polynomials on compact

sets in the complex plane. Arkiv fSr Matematik 8 : 3 17-23 (1969).

4. STOLZE~E~G, G., Uniform appro~m~tion on smooth curves. Acta Math. 115, 185-198 (1966).

5. W~.1~_~R, J., Function rings and Riemann surfaces. Ann. of Math. 67, 45-71 (1958).

6. W ~ R , J., The hull of a curve in C ~. Ann. of Math. 68, 550-561 (1958).

2~4~

Tryckt den 17 november 1970 Uppsala 1970. Almqvist & Wiksells Boktryckeri AB